to prove abc a b c which of the following shows how the asa criterion follows from the defintion of congruence in trems of rigid motions

Answer 1

To show how the ASA (Angle-Side-Angle) criterion follows from the definition of congruence in terms of rigid motions, we can use the following steps:

1. Definition of Congruence via Rigid Motions: Two geometric figures (especially triangles) are congruent if one can be obtained from the other by a series of rigid motions (translations, rotations, reflections). Rigid motions preserve distances and angles.

2. Given Triangles: Consider two triangles, \( \triangle ABC \) and \( \triangle A'B'C' \), where the following are known:

   • Angle \( A \) is congruent to angle \( A' \) ( \( \angle A \cong \angle A' \)).
   • Side \( AB \) is congruent to side \( A'B' \) ( \( AB \cong A'B' \)).
   • Angle \( B \) is congruent to angle \( B' \) ( \( \angle B \cong \angle B' \)).

3. Constructing the Congruence:

   • Start with triangle \( A'B'C' \). Use a rigid motion to place point \( A' \) at point \( A \).
   • Rotate triangle \( A'B'C' \) around point \( A \) until \( \angle A'B'C' \) aligns with \( \angle ABC \) (which is \( \angle A \) in \( \triangle ABC \)).
   • Because \( A'B' = AB \) by congruence of sides, the point \( B' \) will align with point \( B \) after this rotation.
   • Now, \( B \) is already defined, and since \( \angle B' \) will also match due to congruence and the way we rotated triangle \( A'B'C' \), this positioning ensures that point \( C' \) will perfectly align with point \( C \).

4. Conclusion on Congruence: Thus, through the rigid motions used (rotation specifically), we have positioned triangle \( A'B'C' \) such that it exactly overlaps with triangle \( ABC \). This implies the two triangles are congruent by definition. Therefore, the ASA criterion is validated: if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.

This logical flow illustrates that ASA leads to congruence based on the concept of rigid motions, confirming the ASA criterion for triangle congruence.